Is it possible to define a basis for the space of continuous functions?

[Boorglar]

In analogy to vector spaces, can we define a set of "basis functions" from which any continuous function can be written as a (possibly infinite) linear combination of the basis functions?

I know the trigonometric functions 1, sin(nx), cos(nx) can be used for monotonic continuous functions, but not every Fourier series leads to a convergent solution (sinx+sin2x+sin3x+... for example diverges).

Is there a set of functions (not necessarily orthogonal) that spans all continuous functions and does not contain divergent series?

 

[chiro]

Hey Boorglar.

You want to consider a Hilbert-Space and the properties of those along with Banach Spaces in Functional Analysis.

Basically Hilbert-Spaces are continuous in the inner product and Banach Spaces are continuous in the norm: so you can look at these in the context of general functionals and function spaces.

 

[Boorglar]

Yes, I've heard about those spaces. But do they actually provide us with such a basis? I mean, is there an infinite set of known functions that span all the Hilbert space?

 

[pwsnafu]

does not contain divergent series?

Why would this be true? You are asking for $\sum_{i=0}^\infty a_n \, f_n(x)$ where $f_n$ are fixed but the coefficients are arbitrary. We can choose $a_n \rightarrow \infty$ at any rate we want, so there must exist a divergent series (unless all f are zero).

 

[Boorglar]

Why would this be true? You are asking for $\sum_{i=0}^\infty a_n \, f_n(x)$ where $f_n$ are fixed but the coefficients are arbitrary. We can choose $a_n \rightarrow \infty$ at any rate we want, so there must exist a divergent series (unless all f are zero).

hmm yes I guess you're right.

The reason I ask for that is that I was looking for a way to define integration over a space of functions i.e.: let F be a functional, taking a function and returning a number. Then I want to "integrate" this functional with respect to the function argument.

For example: let F[f] = ∫¹₀f(x)dx. Then somehow divide the space of functions into small function intervals (which might perhaps be done if you had a basis) and do a Riemann sum of the values of the functional taken at some arbitrary function in each interval.

If the function space was simply c*e^x where c is a real number, then I could do the integral of the value of F[ce^x] with respect to c from -infinity to +infinity. (this would be a one-dimensional function space)

 

[Stephen Tashi]

For example: let F[f] = ∫¹₀f(x)dx. Then somehow divide the space of functions into small function intervals (which might perhaps be done if you had a basis) and do a Riemann sum of the values of the functional taken at some arbitrary function in each interval.

What is a "function interval"? Are we talking about functions defined on an interval of the real number line? Are "function intervals" going to be subintervals of that interval?

If $f(x) = \sum_{i=0}^\infty c_i g_i(x)$ then the natural way to integrate $f(x)$ over an interval such as [0,1] would be to use an expession $\sum_{i=0}^\infty c_i \int_0^1 g_i(x) dx$ rather than to integrate over subintervals. Are you thinking about breaking [0,1] up into subintervals and using a different basis for each subinterval?

 

[chiro]

How are you defining function? Continuous functions won't be defined analytically in general (although some can be).

The first thing I think you should do is consider a general continuous function in a fixed interval and consider that if it is square integrable (i.e. in L^2) if it has any basis in a Hilbert-Space.

If it has a basis, then consider the properties that this basis must have if you want to go deeper.

I'd look at the first one to "check" that a square integrable function (in L^2(R^n)) over some interval has a basis (any basis) and then you can go from there.

 

[pwsnafu]

hmm yes I guess you're right.

The reason I ask for that is that I was looking for a way to define integration over a space of functions i.e.: let F be a functional, taking a function and returning a number. Then I want to "integrate" this functional with respect to the function argument.

For example: let F[f] = ∫¹₀f(x)dx. Then somehow divide the space of functions into small function intervals (which might perhaps be done if you had a basis) and do a Riemann sum of the values of the functional taken at some arbitrary function in each interval.

If the function space was simply c*e^x where c is a real number, then I could do the integral of the value of F[ce^x] with respect to c from -infinity to +infinity. (this would be a one-dimensional function space)

Ah you want Riemann integration on a function space. Start with Muldowney's paper on the topic (although this is specific to Black-Scholes). There should be a couple of other papers in the literature. I remember Henstock had a paper for an arbitrary metric-measure space.